We’re not starting off with a warmup problem today because I want to jump right into using the software. Manipulating_Circles_on_Geogebra provides a run through of what I demonstrate for students in this opener.
The photo, interactive Graph Applet, shows the checklist of five items I show students while running through this mini-lesson. The Video can also be reviewed by students as a demonstration of what it looks like to run through these steps. During this segment of the lesson I find it’s useful to go through these words to make sure all students know what they mean:
It will help them to better conceptualize the idea that we’re building a tool and learning to use it.
Opening Demo Video URL for sharing with students: http://vimeo.com/62375030
In this section of the lesson I want students to grapple with the project of making an interactive graph applet to model circle equations. I begin by proposing this challenge to students:
Now that you have seen how to make an interactive graph for linear equations, make another one for the equation of a circle. When you’re done, show me, and I’ll give you today’s handout.
I choose this approach because I want to make this knowledge sticky! I like to reiterate repeatedly to students that they know what “m” and “b” do to a line, so now they’re gaining the same sort of knowledge about circles. I ask questions like, "So if m and b are parameters in a linear equation, what sliders will we need to make an applet for circles?" I also use the question, “Why don’t we create sliders for x and y?” to extend conceptual understanding.
The main idea is that I want my students to be able to slide sliders in the circles applet to match the three parameters in a circles equation. Manipulating Circles on Geogebra scaffolds students work towards accomplishing this goal. Nonetheless, I expect that some students will try to type each equation into Geogebra individually. Note that in this way, they’re not using the “interactivity” of the computer applet. So, I emphasize to students that there should only be one equation in their applet, and that they should use sliders to achieve each equation.
I find that it’s satisfying to students to grow the tool as they use it, and, to see its limitations. The default settings in Geogebra will allow students to graph the first circle, but then the second one is more difficult and we have to change the sliders and develop some ways to make this work. Manipulating Circles on Geogebra provides an overview of what this looks like, but I try to be flexible in letting students think of ways to allow students to come up with their own changes to the applet.
If the work with the applet goes well, I give students a Geogebra Circles Problem Solving handout to extend our work with the Equation of Circles. For students who breeze through the first handout, these problems will provide an appropriate challenge. In this handout I leverage geogebra to give students a chance to explore an idea like tangent lines to a circle before they’ve necessarily gained the algebraic skills to access this knowledge.
Video Demo URL for sharing with students: http://vimeo.com/62419253
When there are approximately 30 minutes left in today's lesson, after students are started using their applets to view the graphs of circles, I tell them that at the end of class they’ll have a quiz. On this quiz, I will give them three equations and they will have to graph each circle (see Circles 1 Check In Quiz 1). I note that they now have about 20 minutes to prepare for that quiz by making sure they now how circles work. This focuses their work. I also find it very satisfying to see how students find this overwhelming at first, but the exploration quickly helps them make sense of it, and they feel that sense of accomplishment.
I think it is helpful to students to be strict about time here. The message this sends to students is that once you know how to recognize the center point and the radius from looking at the equation of a circle, it should be easy to make a quick sketch of that circle.
Extension Idea using Geogebra
If you double click on a circle to view its properties, you can then click on “Algebra” and select the general form of the equation. This gives students one entry point for thinking about other forms of equations and completing the square.